Melting and Freezing of Metal Clusters - Semantic Scholar · 2018-11-02 · Andres Aguado ´ 1 and Martin F ... The method of Jarrold and collaborators (45, 73) uses multicollision-induced

PC62CH08-Jarrold ARI 24 February 2011 21:56

Melting and Freezingof Metal ClustersAndres Aguado1 and Martin F. Jarrold2

1Departamento de Fısica Teorica, Universidad de Valladolid, Valladolid 47011, Spain2Chemistry Department, Indiana University, Bloomington, Indiana 47401;email: [email protected], [email protected]

Annu. Rev. Phys. Chem. 2011. 62:151–72

First published online as a Review in Advance onDecember 3, 2010

The Annual Review of Physical Chemistry is online atphyschem.annualreviews.org

This article’s doi:10.1146/annurev-physchem-032210-103454

Copyright c© 2011 by Annual Reviews.All rights reserved

0066-426X/11/0505-0151$20.00

Keywords

thermodynamics, heat capacity, latent heat

Abstract

Recent developments allow heat capacities to be measured for size-selectedclusters isolated in the gas phase. For clusters with tens to hundreds of atoms,the heat capacities determined as a function of temperature usually have asingle peak attributed to a melting transition. The melting temperaturesand latent heats show large size-dependent fluctuations. In some cases, themelting temperatures change by hundreds of degrees with the addition ofa single atom. Theory has played a critical role in understanding the originof the size-dependent fluctuations, and in understanding the properties ofthe liquid-like and solid-like states. In some cases, the heat capacities haveextra features (an additional peak or a dip) that reveal a more complex be-havior than simple melting. In this article we provide a description of themethods used to measure the heat capacities and provide an overview ofthe experimental and theoretical results obtained for sodium and aluminumclusters.

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Dynamic phasecoexistence: asituation in which thesystem fluctuatesbetween being entirelyliquid or entirely solidat the melting/freezingtransition; thisbehavior is found insmall systems

Static phasecoexistence: asituation in which thetwo phases coexist incontact at themelting/freezingtransition; thisbehavior is found inlarge systems

Geometric shell: fora structure for whichthe atoms areorganized in layers,particularly stableclusters can emergewhen a layer isfinished; e.g., foricosahedral packingthere are geometricshell closings at 55 and147 atoms

INTRODUCTION

The melting of a macroscopic crystal is a first-order phase transition. At a given external pressure,the crystal melts at a single temperature at which there is a spike in the heat capacity due to thelatent heat. The reverse process, freezing, often occurs at a temperature significantly below thethermodynamic melting temperature (Tm) because freezing is a nucleated process and the rate ofcritical nucleus formation increases as the temperature is lowered. Melting is usually not nucleated(i.e., crystals melt at T = Tm) because many surfaces premelt, i.e., acquire a thin liquid layer, afew atomic layers thick, as Tm is approached (1–3).

If we reduce the size of a macroscopic object, there is initially no noticeable effect on themelting temperature. The rate of critical nucleus formation is proportional to the volume, so theliquid becomes more deeply supercooled before it freezes. The supercooling of metal dropletswith micrometer diameters was studied by Turnbull and collaborators (4–6) in the early 1950s.

If the size is decreased into the nanometer regime, the surface-to-volume ratio increases sothat the surface energy begins to make an important contribution to the overall energy. Pawlow(7, 8) recognized in 1909 that this leads to a depression of the melting point (because the liquidhas a lower surface energy than the solid) and predicted that the depression should show a 1/rdependency. The melting point depression has been confirmed for many metals (9–13). In thecase of gold (bulk melting point 1,337 K), 3.8-nm diameter particles (approximately 1,700 atoms)melt at ∼1,000 K, and 2.5-nm particles (approximately 500 atoms) melt at ∼500 K (11). If thisbehavior continues, small gold clusters will be liquid at room temperature.

The melting point depression described above results from thermodynamic scaling of thesurface and bulk contributions to the total energy, and the depression is expected to scale smoothlywith particle size. For large particles this is undoubtedly true. But if we continue to decrease theparticle size, at some point its properties must start to show size-dependent fluctuations. In thissize regime, adding a single atom could have a dramatic effect on the melting temperature.

The melting points for large particles mentioned above were obtained by monitoring diffractionpatterns of individual supported particles as a function of temperature. This approach runs intoseveral problems for small particles or clusters. First, the substrate may perturb the properties, andsecond (as we expect size-dependent fluctuations) we need an approach that enables measurementsfor clusters with a known number of atoms. It is only recently that methods have been developedthat allow melting temperatures to be determined for unsupported, single-sized metal clusters asa function of the number of atoms.

Although the small size regime has been a challenge to study experimentally, it has been aprolific area for theoretical studies (14–29). The earlier theoretical studies employed empiricalpotentials, but nevertheless they provided many fascinating predictions, such as that small clustersdisplay a dynamic phase coexistence, instead of the static phase coexistence of macroscopic objects(22, 27, 28).

The first attempt to examine the melting of unsupported metal clusters was by Martin et al. (30).Their method is based on the disappearance of geometric shell structure in the mass spectra of clus-ters as they are heated. However, it does not provide information for single-sized clusters. It wasthe work of Haberland and collaborators (31, 32) that provided the real breakthrough. The mainchallenge with studying melting transitions for unsupported clusters is determining when theymelt. Two basic approaches have been used. The most widely used methods hinge on measuringheat capacities as a function of temperature. The melting temperature is determined from the peakin the heat capacity due to the latent heat. In the other method, ion mobility measurements (33)are used to detect the shape or volume change associated with the melting transition. Experimentalstudies have now been performed for a variety of cluster materials including sodium (31, 32, 34–41),

152 Aguado · Jarrold

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tin (42–44), gallium (45–47), aluminum (48–56), sodium chloride (57), mixed metal clusters [GanAl(58), AlnCu (59)], sodium clusters doped with an oxygen atom (NanO) (60), and water (61). Thesemeasurements have stimulated a number of theoretical studies (49, 62–67). By far the most exten-sive body of work has been performed for sodium and aluminum clusters, and many features ofthe melting of these clusters are now understood thanks to a series of careful theoretical studies.In this article we focus on sodium and aluminum.

Whereas the melting points of large particles are always depressed, this behavior does notalways continue into the small size regime. Small tin and gallium clusters, for example, havemelting temperatures considerably above the bulk value (42, 44, 45). This behavior has beenattributed to the clusters having different bonding than the bulk (68–71).

EXPERIMENTAL METHODS

Most of the methods used to measure melting temperatures of unsupported clusters are basedon measuring the heat capacity as a function of temperature. Calorimetry measurements formacroscopic systems are performed by measuring the temperature as a function of the addedenergy. This approach does not work for unsupported clusters: Although their initial temperaturecan be fixed, there is no way to measure it after a known amount of energy has been added. Thethree methods described here use dissociation to define a reference state that is used to determinethe internal energy change associated with a temperature change (31, 40, 45). They all start withthe preparation of thermalized, size-selected clusters. The need to generate size-selected clusterscalls forth mass spectrometry, which in turn means that the measurements must be performed onions rather than neutral clusters, although both anions and cations can be studied.

The cluster ions are thermalized by passing them through a temperature-regulated regioncontaining a buffer gas. Collisions with the buffer gas bring the clusters to the temperature ofthe walls. Once thermalized, they are in a canonical ensemble. They are then removed to a high-vacuum environment, where they are mass selected and then probed to determine their internalenergy content. In high vacuum, the collision rate is too slow, and the rates of emission andabsorption of radiation are too small, to perturb significantly the internal energy distributionbefore the measurement is completed.

In the method of Haberland and collaborators (31), the size-selected cluster ions are irradiatedwith a laser, undergo multiphoton absorption, and subsequently dissociate. This approach hasbeen applied mainly to study sodium clusters. The dissociation energies of the sodium clusters arearound 1 eV (72), three to four times smaller than the photon energy. In the multiphoton absorp-tion step, enough energy is absorbed to reach the dissociation threshold, and then every additionalphoton leads to the loss of an extra three to four sodium atoms. Thus the fragment ion mass spec-trum shows oscillations with peaks separated by three to four atoms. If more energy is added to thedissociating clusters, either by raising their initial temperature or by changing the photon energy,the peaks in the fragment mass spectrum shift toward smaller product ions. The heat capacity isdeduced by tracking the shift in the fragment mass spectrum as the initial temperature is changed.A drawback of this approach is that the photon energy needs to be much larger than the clusterdissociation energy (which so far has limited this method to weakly bound clusters).

The method of Jarrold and collaborators (45, 73) uses multicollision-induced dissociation.The cluster ions are directed into a collision cell containing helium, and the initial kinetic energyrequired to dissociate 50% of the clusters (KE50%D) is determined. Then the temperature ischanged and KE50%D is measured again. The heat capacity is proportional to the change inKE50%D divided by the change in temperature. The proportionality constant is the fraction ofthe cluster ions’ kinetic energy that is converted into internal energy (74, 75) by collisions with

www.annualreviews.org • Melting and Freezing of Metal Clusters 153

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Global minimum(GM): the lowestenergy structure thatexists on the groundstate of the potentialenergy hypersurface

MD: moleculardynamics

DFT: densityfunctional theory

OF-DFT: orbital-free density functionaltheory

the helium. This is determined from an impulsive collision model. This approach can be appliedto all clusters (strongly bound and weakly adsorbing). The main drawback is that the energies arenot referenced directly to an absolute scale.

In the method of L’Hermite and collaborators (40), the internal energy is probed by deter-mining how many atoms will stick to the clusters. It has been used to investigate the melting ofsodium clusters. Size-selected cluster ions at a well-characterized temperature are passed througha collision cell containing sodium vapor, where additional sodium atoms stick:

Na+n + mNa → Na+

n+m.

Every time a sodium atom sticks to the cluster, the internal energy increases by the dissociationenergy plus the collision energy. As more atoms stick, the internal energy increases to the point atwhich evaporation limits further cluster growth. A cold cluster can accommodate more atoms be-fore reaching the evaporative limit than a hot one. Heat capacities are deduced from measurementsof the number of sodium atoms that stick as a function of temperature and kinetic energy.

THEORETICAL METHODS

A theoretical analysis of cluster melting transitions requires a simulation model and a methodto calculate the cluster energy. The simulation model has to provide information about bothsolid and liquid phases. At zero Kelvin, and dispensing with nuclear quantum effects, the clusteradopts a single solid structure that minimizes the cluster potential energy. Locating this globalminimum (GM) structure is a complex optimization problem for which a variety of algorithms,such as genetic (76) or basin hopping (77), have been developed and reviewed (78). The meltingtransition usually is studied by modeling the heating of the GM structure, either with moleculardynamics (MD) or Monte Carlo techniques (79). MD generates the true dynamics of the clusterand thus provides access to kinetic effects and dynamical properties like time-correlation functions.Monte Carlo is a stochastic method that efficiently samples the appropriate statistical ensemble andprovides faster access to static properties. Both methods suffer from potential broken ergodicity(incomplete sampling) problems (80), which may be alleviated by using multiple histogram (81)or parallel tempering (82) methods, for example. Path-integral techniques (83) can be employedif quantum nuclear effects are important (as in water clusters).

The most accurate cluster energies are provided by ab initio quantum chemistry methods (84),which are limited to very small clusters due to their unfavorable scaling with size. A reliable MDsimulation of cluster melting has to be at least several nanoseconds long, so quantum chemistrymethods typically are not employed. Methods based on density functional theory (DFT) (85)provide a good compromise between accuracy and efficiency, mostly when coupled with pseu-dopotentials (86) to describe the most internal electrons. They are the methods of choice to studythe melting transition of clusters with approximately 100 atoms. Other methods like tight-binding(87) or orbital-free (OF-)DFT (88) can handle up to several thousand atoms and still include anexplicit (although approximate) description of the electronic density. Finally, empirical potentialsare most efficient by dispensing completely with the quantum nature of the problem. If properlyparameterized, they can provide useful semiquantitative information for clusters with more than1,000 atoms or for the more complex case of nanoalloys (89, 90).

MELTING-LIKE TRANSITION IN SODIUM CLUSTERS

Figure 1 shows typical examples of canonical caloric curves for sodium clusters. The meltingtransition is broadened by finite size effects and covers a finite-temperature interval. The derivative


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200 220 240 260 280 300Temperature (K)

–6.29

–6.28

–6.27

–6.26

–6.25

–6.24

En

erg

y (

eV

ato

m–

1)

Na147

255 K

269 K

243 K

q

Na141

Na137

Figure 1Three representative examples of caloric curves for sodium clusters obtained from orbital-free densityfunctional theory simulations. Note the nonmonotonic size dependence of the melting temperatures (arrows).Figure adapted with permission from Reference 110. Copyright (2005) by the American Chemical Society.

Electronic shell: thevalence electrons inmetal clusters arearranged in electronicshells due to angularmomentumrestrictions (like theelectrons in an atom);particularly stableclusters occur atelectronic shellclosings

Premelting: a peak ora shoulder in the heatcapacity at a lowertemperature than themelting transition

of the caloric curve is the canonical heat capacity. The location of the peak in the heat capacity (orthe maximum slope in the caloric curve) is usually taken to be the melting temperature, Tm. Theenergy shift between the solid and liquid branches of the caloric curve (which is equal to the areaunder the heat capacity peak) determines the latent heat, qm. The entropy change for melting,�Sm, then can be determined from Tm = qm/�Sm.

A summary of the measured Tm values is offered in Figure 2. The calorimetric measurementsof Haberland and collaborators (Freiburg group) for Nan

+ clusters with n = 55–360 atoms showa markedly nonmonotonic dependency of Tm with cluster size (32, 37, 39). Maxima in Tm do notcorrelate with electronic shell closings (which dominate most other properties of sodium clusters)(91) or with geometrical shell closings for icosahedral packing. Tm values are depressed by almost40% with respect to the bulk, and moreover the average trend is for Tm to decrease with increasingsize. For clusters with 1,000–10,000 atoms [measurements by Martin and collaborators (30)], Tm

is still depressed by 20%, and a clear extrapolation to the bulk value is still not visible.The measured qm and �Sm values also show a nonmonotonic size dependence. However, most

local maxima in qm and �Sm correlate with known icosahedral shell closings. This observationled Haberland et al. (39) to conclude that qm and �Sm yield more useful information than theirratio Tm and that the size dependence of the melting of sodium clusters is determined mostly bygeometrical effects. The strong correlation between qm and �Sm suggests that the more stableclusters also have stiffer bonds and a more-perfect structural order.

Early computer simulations of the melting-like transition (25, 26, 92–100) achieved only partialsuccess at reproducing and interpreting the measurements for sodium clusters: In particular, thedetailed size oscillations in the melting properties were not reproduced, and many of the simulatedheat capacities showed strong premelting peaks, in marked disagreement with experiment. Thisled Calvo & Spiegelmann (101) to conclude that empirical potentials are qualitatively inadequatefor this problem. Manninen and coworkers (102–105) reported MD simulations at the DFTlevel. Although the simulations were so short that thermal equilibrium was not well established,they reproduced the experimental observation that Tm(Na+

55) > Tm(Na+93). Similar results were

obtained in longer MD simulations by Chacko et al. (106).


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No

rma

lize

d m

elt

ing

te

mp

era

ture

, T m

Approximate inverse cluster radius, N–1/3

1.0

0.9

0.8

0.7

0.6

0.00 0.05 0.10 0.15 0.20 0.25

bcc ? ico

55

142

182299

~104

~103

Martin

Freiburg group

Figure 2Melting temperatures measured for sodium clusters as a function of the inverse cluster radius. The Tm valuesare shown normalized to the bulk limit. The results of Haberland and collaborators (Freiburg group) (32, 37,39) for clusters with 55–360 atoms are shown together with those of Martin and coworkers (30) for clusterswith 103–104 atoms. The red line is a linear regression to the Freiburg group results. For small sizes theclusters are based on icosahedral packing (ico). At some (unknown) critical size, there must be a transitiontoward bulk-like (bcc) structures.

Magic number:a particularly stablecluster size; stabilitycould result from anelectronic shellclosing, a geometricshell closing, or both

Cohesive energy: thetotal binding energy(atomization energy)of a cluster divided bythe number of atoms

Aguado & Lopez (107) reproduced the oscillations in the measured Tm over a broad size range(n = 55–299; see Figure 3) using OF-DFT MD simulations. Clusters with higher Tm values werefound to be more compact and to have a higher surface stability. Noya et al. (108) reportedthe putative GM structures of Nan clusters with up to 380 atoms, obtained with a Murrell-Mottram empirical potential (see Figure 4). The accuracy of this potential was supported byreoptimizations at the OF-DFT level and later confirmed by photoelectron spectroscopy (109) formany sizes [although Ghazi et al. (67) have reported different DFT GM structures for clusters withapproximately 92 atoms]. The theoretical magic numbers (enhanced cohesive energies) showedexcellent agreement with the local maxima in the measured latent heats. For sizes intermediatebetween two icosahedral shell closings, the GM structures undergo unusual twisting distortions,which generate a more stable surface, consisting entirely of {111}-like facets.

The detailed size dependency of melting properties over a narrow size range, n = 135–147,also has been reproduced in DFT simulations (41, 52, 110). Here the cohesive energy has a localmaximum at the icosahedral shell closing (n = 147 atoms), and a large drop at n = 148. Moreover,the cohesive energy of a solid-like cluster with 138 electrons (an electronic shell closing) is notenhanced significantly. The cohesive energies, which correlate with the experimental latent heats(see below), thus are determined mostly by geometric effects for clusters of this size (52). Theentropy of melting also has a local maximum at n = 147. The reduction in �Sm for smaller sizeshas been interpreted in terms of a premelting mechanism involving the diffusive motion of atomicvacancies at the surface (41, 110), in agreement with a combinatorial entropy model advanced byHaberland et al. (39).

Although electronic shell effects are of minor importance for the melting of large sodiumclusters, simulations suggest this is not true for small clusters (111, 112). Here the electron densityis confined to a smaller volume, and the addition or removal of a single electron can induce a


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a

b

14.5

14

13.5

13

280

260

240

220

200

0.005 0.01 0.0151/N

50 100 150 200 250 300N

V/N

(Å

3)

T m (K

)

Experiment

OF-DFT

15

Figure 3(a) Average volume per atom as a function of size. The red dashed line is a linear regression to the data.(b) The melting points determined through orbital-free density functional theory (OF-DFT) moleculardynamics simulations. The simulations reproduce the experimentally observed oscillations in the size rangen = 55–299. Clusters that are more compact melt at a higher temperature. Figure adapted with permissionfrom Reference 107. Copyright (2005) by the American Physical Society.

significant change in the cohesive energies. For example, Ghazi et al. (111) predicted that Tm(Na−39)

is 40 K higher than Tm(Na39) due to the electron shell closing at 40 electrons. These jellium-likeeffects also induce significant shape deformations in both solid and liquid clusters (102–105).

The width of the heat capacity peak generally decreases with increasing cluster size, approach-ing the bulk limit of an infinitely narrow transition. Small clusters, in contrast, melt gradually overa broad temperature interval. The measured specific heat of Na41

+ evolves continuously from asolid value of 3.3kB to a liquid value of approximately 4.1kB (38). The temperature dependency

D5h C5IhC5Ih D3C5 92 T 147 178116 11671 10155

C1D3d C1 Ih D3dC1C1200184 216 232 258 309 357

Figure 4A selection of sodium cluster global-minimum structures that have enhanced stabilities. The structures are labeled by their size andpoint group symmetry. Figure adapted from Reference 108 with kind permission of the European Physical Journal.


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Postmelting: a peakor a shoulder in theheat capacity at ahigher temperaturethan the meltingtransition

Surface layering:in an otherwisedisordered liquid, theatoms or molecules inthe near-surfaceregion are stratifiedinto well-definedlayers parallel to thesurface; the layeringdecays rapidly awayfrom the surface

of the photoabsorption spectrum (113, 114) also suggests a gradual melting process: Cold clus-ters show electronic transitions between discrete molecular orbital levels, whereas for hot liquidclusters the spectrum agrees with a jellium model of confined electrons in a spheroidal well. Thesimulations (115–117) reveal how the liquid phase is approached gradually through isomerizationsbetween different cluster structures, which start at temperatures as low as 30 K and involve the co-operative movement of all the atoms in the cluster. The barriers against these cooperative atomicdisplacements are expected to scale extensively with size, so this mechanism is relevant only forsufficiently small clusters. Some large clusters also may show a broad transition with a small latentheat, if the structure of the solid cluster is amorphous (118). In some cases, such as Na59

+, theamorphous GM structure may be driven by an electronic shell closing, which forces the atoms toadopt a spherical shape (104).

The cluster heat capacities also can show postmelting features, i.e., small peaks or shouldersat T > Tm. For example, the caloric curve of Na147

+ shows such a feature at 40 K above Tm

(41). This postmelting feature has been attributed to the presence of surface layering in the liquidcluster (see Figure 5), a property shared by extended liquid metal surfaces. The surface layering issize dependent. For Na147

+, which has a nonmelting surface (41), the layering is sufficiently strongto effectively confine the 13 innermost atoms in a solid-like phase even above Tm; the cluster corecannot melt completely until 40 K above Tm. Na139

+, in contrast, has some atomic vacancies at thesurface, which facilitates radial atomic diffusion; the layering is correspondingly not so strong andno postmelting anomaly is observed. Postmelting anomalies have been observed in simulations ofvery small clusters such as Na25 (119). In this case they are the cluster analog of a liquid-liquidphase transition between two liquid phases with different average densities.

There is an interesting theoretical prediction concerning the melting-like transition in clus-ters and the nonequivalence of statistical ensembles for finite, nonextensive systems. In the

Normalized radial distance (r/rcluster)

Ato

mic

de

nsi

ty ×

10

–4 (

a.u

.)

Na (N = 139)

0.2 0.4 0.6 0.8 1 1.20

5

10

15

Na (N = 147)

LJ (N = 147)

Figure 5Calculated radial atomic densities of Na147 and Na139 as a function of normalized distance, where rcluster isthe cluster radius and r measures the distance from the cluster’s center of mass. The results correspond to atemperature that is above Tm but below the postmelting temperature; i.e., the core of Na147 is still solid. Thelayering has a larger amplitude for Na147: The first minimum even touches the r axis, demonstrating that thecentral atom does not diffuse at this temperature, a clear indicator of postmelting. A 147-atom Lennard-Jones (LJ) cluster just above Tm (blue dashed line), conversely, shows no layering at all. Figure adapted withpermission from Reference 41. Copyright (2009) by the American Physical Society.


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Energy (eV)

He

at

cap

aci

ty (

k B)

–20

–15

–10

–5

0

5

10

15

2 4 6 8 10 12 14 16

160

180

200

7 8 9 5 × 106 1 × 107120

160

200

240

T (K

)

Time (fs)0

Figure 6Microcanonical heat capacity of Na147 obtained from parallel tempering Monte Carlo simulations (122).The lower-left inset shows the microcanonical and canonical ( gray dashed line) caloric curves. Thelower-right inset shows the time evolution of the kinetic temperature at an energy within the dynamicalcoexistence range. Figure adapted with permission from Reference 122. Copyright (2008) by theEurophysics Letters Association.

microcanonical ensemble, the basic quantity (from which all the thermodynamic information maybe extracted) is the entropy, S = kB ln �. In finite systems S is a nonextensive quantity and may de-velop a dent with an inverted curvature for energies within the dynamical coexistence range. If thisis the case, the microcanonical caloric curve acquires a negative slope (or undergoes backbending)and the heat capacity becomes negative at melting. The canonical energy distribution becomesbimodal, with a deep minimum between the peaks corresponding to solid and liquid phases. TheS dent implies a reduced number of states at the melting energies: States with solid and liquidphases in contact (static coexistence) are shifted to higher energies because of the interfacial freeenergy (which is not negligible in a finite system), and dynamical coexistence occurs instead. Thelatent heat must be larger than typical excitation energies under coexistence conditions in orderto observe a negative heat capacity (120). A negative heat capacity has been inferred for Na147

+

from the experiments by Schmidt et al. (35), and it has been reproduced in simulations (121, 122)(see Figure 6). Michaelian and coworkers (123) have argued that the negative heat capacity is anartifact due to broken ergodicity (trapping in a restricted region of phase space). However, thereis sufficient evidence and quite a general consensus (122, 124) that the S dent is a real feature ofnonextensive systems.

MELTING-LIKE TRANSITION IN ALUMINUM CLUSTERS

Heat capacities have been measured for aluminum cluster cations (16–128 atoms) and anions(35, 70) using multicollision-induced dissociation. Figure 7 shows heat capacities recorded foraluminum cluster cations with 40, 42, 43, and 37 atoms (50). All clusters in Figure 7 show a singlepeak, which is attributed to a melting transition. As noted above, the melting of small clusters isthought to involve a dynamic coexistence between the solid-like and liquid-like states, in whichthe entire cluster fluctuates between solid and liquid. Dynamic phase coexistence can be treated asa two-state transition (equilibrium) between the solid-like and liquid-like states (125). This leads


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Temperature (K)0 300 600 900 1,200

60

99

161

161

Al40+

Al42+

Al43+

Al37+

Figure 7Plot of the heat capacities for aluminum cluster cations with 40, 42, 43, and 37 atoms (50). The blue squaresshow the measured heat capacities, and the red lines show fits using a two-state melting model that invokesan equilibrium between solid-like and liquid-like clusters. The numbers on the right-hand side of the figureshow the latent heat associated with each melting transition in kilojoules per mole. Al40

+, Al42+, and Al43

+have approximately the same melting temperatures while the latent heat increases with the cluster size. Themelting transition becomes narrower as the latent heat increases, as expected for a two-state transition. Al37

+has the same latent heat as Al43

+, but it melts at a significantly higher temperature. The transition for Al37+

is broader than for Al43+, in agreement with the prediction of the two-state model.

to a straightforward prediction for the width and height of the peak in the heat capacity (Figure 7)and its dependency on the latent heat and Tm. The peaks get broader as the latent heat decreasesand as Tm increases because the solid-like and liquid-like states coexist over a broader temperaturerange.

The heat capacities of some clusters show two peaks. Results for Al115+ are illustrated in

Figure 8. Similar behavior was found for Al116+ and Al117

+ (54). These are the only clustersfor which two clearly resolved peaks were observed. For some other clusters, the heat capacitypeak is bimodal or broadened, indicating that melting does not occur through a simple two-statetransition.

There are many examples in which two heat capacity peaks have been observed in simulations(71, 95, 97, 101, 117, 118, 126–128). This behavior is usually called premelting when the low-temperature peak is the smaller of the two and postmelting for the reverse situation. However,multiple peaks are rare in experimental studies. As noted above, a high-temperature shoulder on


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Temperature (K)200 400 600 800 1,000

C/3

Nk B

Al115+

Unannealed

Annealed(523 K)

Annealed(773 K)

Figure 8Heat capacities recorded for Al115

+. The upper plot shows results for unannealed Al115+. The middle plot is

for an annealing temperature of 523 K (between the two peaks in the upper plot). The lower plot is for anannealing temperature of 773 K (above both peaks). The arrows on the upper plot show the two annealingtemperatures. Figure adapted with permission from Reference 54. Copyright (2009) by the AmericanPhysical Society.

the heat capacity peak for Na147+ is attributed to liquid layering (41). For aluminum, there are

several examples of both premelting (Al51+, Al52

+, Al115+, Al116

+, and Al117+) (48, 51, 54) and

postmelting (Al61+ and Al83

+) (50, 51).Excluding electronic effects, there are three main causes for a second feature in the heat capacity

plot: (a) partial melting, in which the surface or some other part of the cluster melts at a lowertemperature than the rest; (b) two structural isomers that melt at different temperatures; and (c) asolid-to-solid transition followed by melting.

Structural transitions have been seen to precede melting in a number of recent simulations (129–136). For example, in simulations of palladium clusters, transitions from both fcc and decahedralground states to icosahedral structures precede melting (136). Solid-solid structural transitions arefrequently found to precede melting for Lennard-Jones clusters, resulting in two well-resolvedpeaks in the heat capacity (137–143).

Annealing studies were performed for Al115+, Al116

+, and Al117+ to shed light on the origin of the

two peaks (54). In these experiments the clusters were heated and held at an elevated temperaturebefore having their temperature set for the heat capacity measurements. Al115

+ was annealed to


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523 K and 773 K (see arrows in Figure 8). In both cases the low-temperature peak disappearedand the high-temperature peak was unaffected. Based on these results, the low-temperature peakis attributed to a structural transition (54) and the high-temperature peak to the melting of thehigher-enthalpy structure generated in the structural transition. The structural transition involvesa superheated solid (it does not occur under equilibrium conditions).

A peak in the heat capacity indicates a transition to a higher enthalpy, but entropically pre-ferred state. For some clusters with approximately 60 and 80 atoms, there is a dip in the heatcapacities at a slightly lower temperature than the peak (51). A dip in the heat capacity indicatesa transition to a lower enthalpy state. For example, a dip could result from a structural transitionfrom a poorly formed (amorphous) geometry generated during cluster growth to a lower-energystructure. If this explanation is correct, the dips should disappear when the clusters are annealed,but for many clusters the dips persist. The behavior of the dips can be reproduced using a simplekinetic model of melting and freezing in a system of one liquid-like and two solid-like states withdifferent melting temperatures and latent heats. The dips result from freezing into a high-energygeometry and then annealing into the thermodynamically preferred solid. The thermodynami-cally preferred solid has the higher freezing temperature. However, the liquid can bypass freezinginto the thermodynamically preferred solid (at high cooling rates) if the higher-energy geometryhas a larger freezing rate (just like glass formation in the macroscopic world).

Figure 9 shows melting temperatures and latent heats from Al25+ (the smallest cluster to show

a heat capacity peak) to Al128+. Most clusters have melting temperatures below the bulk value,

a

b

20 30 40 50 60 70 80 90 100 110 120 130

1,100

1,000

900

800

700

600

500

400

300

Me

ltin

g t

em

pe

ratu

re (

K)

7

6

5

4

3

2

1

0

20 30 40 50 60 70 80 90 100 110 120 130

La

ten

t h

ea

t (e

V)

Cluster size (atoms)

Bulk

MP

Figure 9Melting temperatures (a) and latent heats (b) for aluminum cluster cations with 25 to 128 atoms.


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but the melting temperatures do not drop systematically with decreasing cluster size. There arelarge fluctuations in the melting temperatures for clusters with less than 90 atoms, and smootheroscillations for n > 90. The latent heats also show large fluctuations that are not correlated withthe melting temperatures. Efforts to understand the fluctuations in melting temperatures andlatent heats using empirical potentials have been largely unsuccessful (62–64).

Fluctuations in the latent heats are correlated with fluctuations in the cohesive energies of solidclusters (52). Usually, dissociation energies for metal clusters are measured by adding energy until

3.05

3.00

2.95

2.90

2.85

3.0

2.5

2.0

1.5

1.0

0.5

0.030 40 50 60 70 80

30 40 50 60 70 80

Co

he

siv

e e

ne

rgy

(e

V a

tom

–1)

La

ten

t h

ea

t (e

V)

Cluster size (atoms)

43,44

57

37 39

Figure 10Plots showing the correlation between the latent heats and the cohesive energies of solid clusters (52). Thepurple solid circles represent the cohesive energies of liquid clusters. The blue solid circles represent thecohesive energies of solid clusters (which are determined from the cohesive energies of liquid clusters andthe latent heats). The red open circles show cohesive energies for the lowest energy structures found in anextensive geometry search with density functional theory. The measured and calculated cohesive energiesare in good agreement except for a few clusters around Al57

+ and for clusters larger than 71 atoms. For theseclusters, the calculated cohesive energy is smaller than the measured value, and it is likely that the lowestenergy structure has not been found in the calculations.


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the cluster dissociates (72, 144–147). For clusters with more than a few atoms, an excess energy(i.e., beyond the dissociation energy) is required for dissociation to occur on the experimentaltimescale. Then, the dissociation energy is deduced using a statistical model to account for theexcess energy. For clusters that melt, this procedure provides the dissociation energy for the liquidcluster, because dissociation occurs from the liquid state. Figure 10 shows cohesive energiesdetermined using this sort of approach for liquid aluminum cluster cations (52). As expected, thecohesive energies of the liquid clusters change smoothly with cluster size, suggesting that electronshell closing effects are not significant in liquid aluminum clusters. The cohesive energies of thesolid clusters are given by

CS(n) ≈ L(n) + CL(n)n

,

where CL(n) is the cohesive energy of the n atom liquid cluster, and L(n) is the latent heat. As CL(n)does not show significant size-dependent fluctuations, fluctuations in the cohesive energies of thesolid-like clusters result almost entirely from fluctuations in the latent heats (52).

Measured cohesive energies for the solid clusters also are shown in Figure 10. There aresignificant fluctuations in the cohesive energies of the solid clusters. Figure 10 additionally showscohesive energies for the lowest energy structures found using an extensive search with DFT. Fourgroups of low-energy structures were found: distorted decahedral fragments, fcc fragments, fccfragments with stacking faults, and disordered structures (see Figure 11 for examples) (52). The

Distorteddecahedralfragments

(ddf)

fcc

fcc withstacking faults

(sf)

Disorderedpolytetrahedral

(dis)

36

63

41

46

55

75

72

65

Figure 11Examples of the four types of geometry found for aluminum clusters in the density functional theorycalculations.


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400

600

800

1,000

30 40 50 60 70Cluster size (atoms)

Me

ltin

gte

mp

era

ture

(K)

ddf+ dis sf

sf+ dis ddf (sf + fcc) dis sf + fcc

Figure 12Comparison of the melting temperatures for aluminum cluster cations (blue open circles) and anions (red solidcircles). The gray notations along the top of the plot identify the lowest energy structure from DFTcalculations associated with each size range: ddf, distorted decahedral fragments; fcc, fcc fragments; sf, fccfragments with stacking faults; and dis, disordered structures. Large changes in the melting temperatures arecorrelated with structural changes. Figure adapted with permission from Reference 55. Copyright (2009) bythe American Physical Society.

measured and calculated cohesive energies (Figure 10) are correlated strongly for most clustersizes. According to the calculations, the variations in the cohesive energies (and the latent heats)result from a combination of geometric and electronic shell effects. For some clusters an electronicshell closing is responsible for the enhanced cohesive energy and latent heat (e.g., n = 37), whereasfor others (e.g., n = 44) a structural shell closing is the cause.

Figure 12 compares melting temperatures determined for aluminum cluster anions and cations(55). The melting temperatures for the anions and cations show the same general features, butthose for the anions are shifted to slightly smaller sizes. The large variations in the melting tem-peratures are correlated with changes in the structure of the clusters. For example, the maximumin the melting temperatures around 66 atoms is associated with the clusters adopting disorderedgeometries that result from a spherical electronic shell closing with 198 valence electrons. Theelectronic shell closing forces the cluster to adopt a near-spherical geometry, which has much lesslocal order than the other geometries. As a consequence, the entropy change on melting is smaller,and this causes an elevated melting temperature according to Tm = qm/�Sm. The downward shiftin the anion features in Figure 12 is probably an electronic effect (the anion has two more valenceelectrons than the cation).

FUTURE WORK

The studies described here barely scratch the surface of cluster melting and freezing, and there aremany interesting and important questions that remain to be addressed. For example, it is not knownat what size the liquid and solid phases of a cluster start to coexist in contact; virtually nothingis known about the kinetics of cluster melting and freezing transitions and how the rates changewith size. The properties of the highly confined liquid clusters remain a largely unexplored area,


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although studies of the chemical properties of liquid aluminum clusters show that large changesin reactivity can be associated with the melting transition (148, 149). The interplay betweenstructure, dynamics, phase, and reactivity has connections to catalysis. Finally, we mention thevast and virtually unexplored area of bimetallic clusters and the potential for tailoring propertiesat the nanometer length scale with alloys and core-shell arrangements.

SUMMARY POINTS

1. Metal clusters with fewer than 100 atoms show melting/freezing transitions.

2. The melting temperatures and latent heats show large size-dependent fluctuations.

3. In some cases, the addition of a single atom can change the melting temperature byhundreds of degrees.

4. The size-dependent variations in the latent heats are correlated with variations in thecohesive energies.

5. The size-dependent fluctuations in the melting temperatures are related mainly to struc-tural changes.

6. In some cases the liquid clusters show evidence for layering (in which the atoms in theliquid are organized in layers).

7. The microcanonical heat capacity can become negative on melting.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings thatmight be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

We gratefully acknowledge the support of the Spanish “Ministerio de Ciencia e Innovacion,”the European Regional Development Fund, and “Junta de Castilla y Leon” (project numbersFIS2008-02490/FIS and GR120) and the support of the U.S. National Science Foundation undergrant 0911462.

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PC62FrontMatter ARI 18 February 2011 17:56

Annual Review ofPhysical Chemistry

Volume 62, 2011 Contents

Laboring in the Vineyard of Physical ChemistryBenjamin Widom � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

The Ultrafast Pathway of Photon-Induced ElectrocyclicRing-Opening Reactions: The Case of 1,3-CyclohexadieneSanghamitra Deb and Peter M. Weber � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �19

Coarse-Grained (Multiscale) Simulations in Studies of Biophysicaland Chemical SystemsShina C.L. Kamerlin, Spyridon Vicatos, Anatoly Dryga, and Arieh Warshel � � � � � � � � � � � �41

Dynamics of Nanoconfined Supercooled LiquidsR. Richert � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �65

Ionic Liquids: Structure and Photochemical ReactionsEdward W. Castner Jr., Claudio J. Margulis, Mark Maroncelli,

and James F. Wishart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �85

Theoretical Study of Negative Molecular IonsJack Simons � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 107

Theoretical and Computational Protein DesignIlan Samish, Christopher M. MacDermaid, Jose Manuel Perez-Aguilar,

and Jeffery G. Saven � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 129

Melting and Freezing of Metal ClustersAndres Aguado and Martin F. Jarrold � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 151

Astronomical ChemistryWilliam Klemperer � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 173

Simulating Chemistry Using Quantum ComputersIvan Kassal, James D. Whitfield, Alejandro Perdomo-Ortiz, Man-Hong Yung,

and Alan Aspuru-Guzik � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 185

Multiresonant Coherent Multidimensional SpectroscopyJohn C. Wright � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 209

Probing Free-Energy Surfaces with Differential Scanning CalorimetryJose M. Sanchez-Ruiz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 231

viii

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Role of Solvation Effects in Protein Denaturation: FromThermodynamics to Single Molecules and BackJeremy L. England and Gilad Haran � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 257

Solid-State NMR Studies of Amyloid Fibril StructureRobert Tycko � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 279

Cooperativity, Local-Nonlocal Coupling, and Nonnative Interactions:Principles of Protein Folding from Coarse-Grained ModelsHue Sun Chan, Zhuqing Zhang, Stefan Wallin, and Zhirong Liu � � � � � � � � � � � � � � � � � � � � � 301

Hydrated Acid ClustersKenneth R. Leopold � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 327

Developments in Laboratory Studies of Gas-Phase Reactions forAtmospheric Chemistry with Applications to Isoprene Oxidationand Carbonyl ChemistryPaul W. Seakins and Mark A. Blitz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 351

Bonding in Beryllium ClustersMichael C. Heaven, Jeremy M. Merritt, and Vladimir E. Bondybey � � � � � � � � � � � � � � � � � � � 375

Reorientation and Allied Dynamics in Water and Aqueous SolutionsDamien Laage, Guillaume Stirnemann, Fabio Sterpone, Rossend Rey,

and James T. Hynes � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 395

Detecting Nanodomains in Living Cell Membrane by FluorescenceCorrelation SpectroscopyHai-Tao He and Didier Marguet � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 417

Toward a Molecular Theory of Early and Late Events in Monomerto Amyloid Fibril FormationJohn E. Straub and D. Thirumalai � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 437

The Density Matrix Renormalization Group in Quantum ChemistryGarnet Kin-Lic Chan and Sandeep Sharma � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 465

Thermodynamics and Mechanics of Membrane Curvature Generationand Sensing by Proteins and LipidsTobias Baumgart, Benjamin R. Capraro, Chen Zhu, and Sovan L. Das � � � � � � � � � � � � � � � 483

Coherent Nonlinear Optical Imaging: Beyond FluorescenceMicroscopyWei Min, Christian W. Freudiger, Sijia Lu, and X. Sunney Xie � � � � � � � � � � � � � � � � � � � � � � � 507

Roaming RadicalsJoel M. Bowman and Benjamin C. Shepler � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 531

Coarse-Grained Simulations of Macromolecules:From DNA to NanocompositesJuan J. de Pablo � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 555

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New Developments in the Physical Chemistry of Shock CompressionDana D. Dlott � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 575

Solvation Dynamics and Proton Transfer in Nanoconfined LiquidsWard H. Thompson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 599

Nonadiabatic Events and Conical IntersectionsSpiridoula Matsika and Pascal Krause � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 621

Lessons in Fluctuation Correlation SpectroscopyMichelle A. Digman and Enrico Gratton � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 645

Indexes

Cumulative Index of Contributing Authors, Volumes 58–62 � � � � � � � � � � � � � � � � � � � � � � � � � � � 669

Cumulative Index of Chapter Titles, Volumes 58–62 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 672

Errata

An online log of corrections to Annual Review of Physical Chemistry articles may befound at http://physchem.annualreviews.org/errata.shtml

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